A structural classification of candidate oscillators and multistationary systems

Molecular systems are uncertain: the variability of reaction parameters and the presence of unknown interactions can weaken the predictive capacity of solid mathematical models. However, strong conclusions on the admissible dynamic behaviors of a model can often be achieved without detailed knowledge of its specific parameters. In particular, starting with Thomas' conjectures, loop-based criteria have been largely used to characterize oscillatory and multistationary dynamic outcomes in systems with a sign definite Jacobian.\n\nWe build on the rich literature focused on the identification of potential oscillatory and multistationary behaviors based on parameter-free criteria. We propose a classification for sign-definite non autocatalytic biological networks which summarize several existing results in the literature, adding new results when necessary. We define candidate oscillators and multistationary systems based on their admissible transitions to instability. We introduce four categories: strong/weak candidate oscillatory/multistationary systems, which correspond to networks in which all/some of the existing feedback loops are negative/positive. We provide necessary and sufficient conditions characterizing strong and weak candidate oscillators and multistationary systems based on the exclusive or simultaneous presence of positive and negative loops in their linearized dynamics. We also consider the case in which the overall system is the connection of several stable aggregate monotone components, providing conditions in terms of positive/negative loops in a suitable network with aggregate monotone systems as nodes.\n\nMost realistic examples of biological networks fall in the gray area of systems in which both positive and negative cycles are present: therefore, both oscillatory and bistable behavior are in principle possible. Native systems with a large number of components are often interconnections of monotone modules, where negative/positive loops among modules characterize oscillatory and bistable behaviors, in agreement with our results. Finally, we note that many canonical example circuits exhibiting oscillations or bistability fall in the categories of strong candidate oscillators/multistationary systems.